for my car, as an example, goes like this:
1) the evaporator sensor (#5) cycles between 36F and 43F with the engine at 1500 rpm.
2) the refrigerant pressure (#6) cycles between 8 to about 14 or so. Initially, it will start from about 5 or 6 and then build as the A/C operates. Also if it's a warm/hot day, the readings will vary abit.

If your readouts are the same as mine, then there's nothing wrong with your A/C! It could be the pushbutton's contact on the panel.

If #5 reads low, that might explain why the A/C did not turn on. Not enough pressure to activate the switch.